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# 1 (3) - b u +ICJ/‘LJ Class Test 1 ECE502 — Analysis of...

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Unformatted text preview: b u +ICJ/‘LJ Class Test 1 ECE502 — Analysis of Probabilistic Signals and Systems (Fall 2009) Instructor: Professor A. M. Wygljnski October 5, 2009 06:00pm — 06:50pm Instruct ions 0 The duration of this class test is 50 minutes. 0 This class test consists of three (3) questions. each of which has a different point value. a This class test is closed book. a One letter size sheet with handwritten notes on both sides is permitted. a A photocopy of the inner cover of the course textbook is permitted. 9 Calculators are permitted. 0 Please write neatlyr and legibly. - Each point in this class test corresponds to one minute of doing the test. Therefore, this test is out. of 50 points. Budget. your time accordingly. :- Attempt all questions: clearly show all your work and not just the ﬁnal answer. - Collaborating with other students during the class test is strictly prohibited. Hand in all material to the instructor at the end of 50 minutes. I (v'oori luck! m/so :~) WW Last Name First Name .Student ID Mailbox Question 1: Multiple Random Variables [25 Points] In H. Mass of ‘20 students, suppose that the number of coins in each student’s pocket can be deﬁned as a uniform random variable between zero and twenty five. Asstune that the number of coins in the packets of the different students is independent. (21) 4 points: Deﬁning the mnnber of coins centajned in the pocket of student k as the random Variable Xk, express the probability P({X1 Z n} ﬂ'({X2 2 71} ﬂ - - - ﬂ{Xm Z n}) in terms of P(,\'i.2n),k:1.21.”,20. (b) 4 points: Find the probability that no student has fewer than ﬁve coins in his/her pocket. HINT: Use the result in part (a). (c) 5 points: What it the probability that at least one student has at least 19 coins in his / her pocket? ((1) 6 points: W hat is the probability that only student 1:: has catectly 19 coins in his/her pocket? (e) 6 points: What. is the probability that only one student has exactly 19 coins in his/her pocket? HINT: Use the result from pent and generalize. Question 1 (Continued) Question 1 (Continued) Question 2: Conditional Probability [15 Points] The Acme Probability Products Company has been tasked to construct a random number generator for a client. The client specifically requests a random number generator possessing an output statisti- cally deﬁned as a Poisson probability mass function (pnﬁ). Moreover, the client wants this generator to be capable of using one of three possible A parameters: a, b, or (2. Consequently. the Acme Probability Products Company proposed the design shown in Figure l, where Y is the sum of the random outputs of a bank of random number generators. and X is used to select the desired random number generator. When a random number generator is not selected, it only generates a steady stream of zeros. Thus, when X = 1 the output Y ~ Poisson (a), when X : 2 the outth i" ~ Poisson(b), and when X = 3 the output Y «1 Poisson(c). Note that X consists only of the equally likely values 1. 2, and 3. < i ___.Flk t. v *o i Q. E a: I 3 i E. a e i _.—si: : 3 Poisson(c) __ ates Switch Bank of Random Number Generators Figure 1: Schematic of a random number generator bank employing a selector switch. (a) 3 points: What are the mathematical expressions for the conditional probabilities P(Y = an = 1). my = n|X = 2), and PO” = 10le = 3)? (b) 6 points: What is the probability that the output Y is a value less than or equal to 1, Le... PO" 5 1)? (c) 6 points: What is the probability that the input to the selector switch is equal to 2 given that th. ' is a value less than or equal to 1, 216., P[X = 2|Y 5 1)? Question 2 (Continued) Question 3: Moment Generating Functions [10 Points] Suppose the moment. generating function of a, random variable X is deﬁned by: Mxl-Sl = Elﬁsxl Z Z Bmm'lmil (ll 1' wlwre- px (:15) is the probability nun-58 function of X. (a) 47 points: Express Equation (1) in terms of E[X"]. HINT: The series expansion of an exponential may be useful. (b) 3 points: Suppose E[X“‘] = 0.8, A“. = 1,2, . . ., ﬁnd the moment generating function MAS). (c) 3 points: Find P(X = (J) and P(X = l). HINT: Part (b) can be helpful. @wnmmeezjg l) Question 3 (Continued) Question 3 (Continued) 1f] ...
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## This note was uploaded on 09/27/2011 for the course ECE 2010 taught by Professor Stephenbitar during the Spring '09 term at WPI.

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1 (3) - b u +ICJ/‘LJ Class Test 1 ECE502 — Analysis of...

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